Families of Explicit Isogenies of Hyperelliptic Jacobians

Benjamin Smith 1
1 TANC - Algorithmic number theory for cryptology
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France, Polytechnique - X, CNRS - Centre National de la Recherche Scientifique : UMR7161
Abstract : We construct three-dimensional families of hyperelliptic curves of genus 6, 12, and 14, two-dimensional families of hyperelliptic curves of genus 3, 6, 7, 10, 20, and 30, and one-dimensional families of hyperelliptic curves of genus 5, 10 and 15, all of which are equipped with an an explicit isogeny from their Jacobian to another hyperelliptic Jacobian. We show that the Jacobians are generically absolutely simple, and describe the kernels of the isogenies. The families are derived from Cassou--Noguès and Couveignes' explicit classification of pairs $(f,g)$ of polynomials such that $f(x_1) - g(x_2)$ is reducible.
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Conference papers
David Kohel and Robert Rolland. Arithmetic, Geometry, Cryptography and Coding Theory 2009, Mar 2009, Luminy, France. American Mathematical Society, 521, pp.121-144, 2010, Contemporary Mathematics
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Benjamin Smith. Families of Explicit Isogenies of Hyperelliptic Jacobians. David Kohel and Robert Rolland. Arithmetic, Geometry, Cryptography and Coding Theory 2009, Mar 2009, Luminy, France. American Mathematical Society, 521, pp.121-144, 2010, Contemporary Mathematics. 〈inria-00420605〉

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